What is the (higher order) time derivative of centripetal acceleration?

Just using basic dimensional analysis, it appears the time derivative of centripetal acceleration is ## \vec{r} \omega^3 ##, but this intuitive guess would also extend to higher order time derivatives, no? Implying:

## \frac {d^n \vec{r}}{dt^n} = \vec{r} \omega^n ##

It seems to follow from the general result shown in Thorton/Marion pg 390 (attached) when considering rotating bodies in a fixed frame. I assume ## \vec{Q} ## is any vector, even ones that are the result of a higher order time derivative of an initial vector. The concept of finite infinite-time derivative just seems like an odd concept to me when considering real objects, but I guess the geometry of the situation allows it. But to confirm, is anything posted here incorrect?

Just using basic dimensional analysis, it appears the time derivative of centripetal acceleration is r⃗ ω3r→ω3 \vec{r} \omega^3 , but this intuitive guess would also extend to higher order time derivatives, no? Implying:

Dimensional analysis allows that as a possibility, but it does not make it true. If we start with the scalar form ##r\omega^2## we see the time derivative is ##r\dot\omega^2+2r\omega\dot{\omega}##.